Use the graphs of $f$ and $g$ to find each value.

A graph of a curve and a line. The x axis ranges from negative 2 to 6 and the y axis ranges from negative 2 to 4. The line for g passes through the ordered pairs 2 comma negative 2, 3 comma 0, 6 comma 3. The curve for f passes through negative 1 comma 4, 1 comma 0, and 3 comma 4. 

a.  $f(g(6))=$ 

b.  $g(f(-1))=$  

c.  $f(f(2))=$  

d.  $g(g(3))=$  


To solve these problems, we'll evaluate the given function values based on the provided graph description of the functions $ f $ and $ g $. Let's go step by step:

### a. $ f(g(6)) $

1. **Find $ g(6) $:**  
   According to the description, the line $ g $ passes through the point $ (6, 3) $. So, $ g(6) = 3 $.

2. **Find $ f(3) $:**  
   From the description of the curve for $ f $, it passes through the point $ (3, 4) $. Hence, $ f(3) = 4 $.

Thus, $ f(g(6)) = f(3) = 4 $.

### b. $ g(f(-1)) $

1. **Find $ f(-1) $:**  
   From the graph description, the function $ f $ passes through $ (-1, 4) $, so $ f(-1) = 4 $.

2. **Find $ g(4) $:**  
   Since there is no direct information about $ g(4) $ in the description, we need to assume based on the line's behavior or additional points if available. As no specific $ g(4) $ point is given, we may infer that this value could lie between existing points if interpolation is required. However, without exact interpolation details, this part is incomplete.

### c. $ f(f(2)) $

1. **Find $ f(2) $:**  
   The description does not provide the exact value for $ f(2) $. However, we know that the curve passes through points $ (1, 0) $ and $ (3, 4) $, so we can assume that $ f(2) $ lies between those values. Given the curve's trend, it seems that $ f(2) = 2 $ might be a reasonable interpolation.

2. **Find $ f(2) $:**  
   From the assumption above, $ f(2) = 2 $.

Thus, $ f(f(2)) = f(2) = 2 $.

### d. $ g(g(3)) $

1. **Find $ g(3) $:**  
   From the graph description, $ g(3) = 0 $.

2. **Find $ g(0) $:**  
   From the graph, the value of $ g(0) $ is not directly given, but based on the points $ (2, -2) $, $ (3, 0) $, and extrapolating the linear trend, we can reasonably assume that $ g(0) = -1 $.

Thus, $ g(g(3)) = g(0) = -1 $.

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### Final Answers:
a. $ f(g(6)) = 4 $  
b. $ g(f(-1)) = $ *Incomplete, requires $ g(4) $*  
c. $ f(f(2)) = 2 $  
d. $ g(g(3)) = -1 $

Would you like further clarification on any part, or would you like me to provide more detail on how interpolation works for these problems?

Here are 5 questions related to this concept:
1. How do you interpret composite functions graphically?
2. What are typical methods for estimating values between points on a graph?
3. How does linear interpolation work for line graphs like $ g(x) $?
4. How do you calculate the value of a function at non-given points on a graph?
5. What strategies can you use to solve composite functions when some values are unknown?

**Tip:** When working with graphs, estimating values between known points using trends like linear or curve fitting is a useful strategy when exact values are not provided.

Use the graphs of $f$ and $g$ to find each value.
A graph of a curve and a line. The x axis ranges from negative 2 to 6 and the y axis ranges from negative 2 to 4. The line for g passes through the ordered pairs 2 comma negative 2, 3 comma 0, 6 comma 3. The curve for f passes through negative 1 comma 4, 1 comma 0, and 3 comma 4.
a. $f(g(6))$
b. $g(f(-1))$
c. $f(f(2))$
d. $g(g(3))$

Learn how to solve composite function problems using the graphs of f and g. Evaluate values like f(g(6)), g(f(-1)), f(f(2)), and g(g(3)) by interpreting the graph of a line and a curve. This guide covers graph reading and interpolation methods, perfect for Grades 9-12.

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